Consider a Riemannian manifold $\mathcal{M}$ modeled on a (possibly infinitely dimensional) Hilbert space. Suppose that $\{p_i\} \subset \mathcal{M}$ and $p_i \to p$ .
We say that a sequence of covectors $v^*_i \in T^*_{p_i}\mathcal{M}$ is weakly converges to $v \in T_p^*\mathcal{M} $ if for every smooth function $f$ on $\mathcal{M}$ we have $$\langle v^*_i, df(p_i) \rangle \to \langle v^*, df(p) \rangle. $$
Question: Is it true that if $\{v_i^*\}$ is bounded then it has a weakly convergent subsequence?